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The pressure and density of a diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes adiabatically from $(P, d)$ to $\left(P^{\prime}, d^{\prime}\right)$. If $\frac{d^{\prime}}{d}=32$. Then, $\frac{P^{\prime}}{P}$ is
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$128$
$\gamma=\frac{7}{5}$
For adiabatic process $P V^\gamma=$ constant where,
$V=$ volume
$\therefore \quad P d^{-7}=$ constant
$\left(\because v \propto \frac{1}{d}\right)$
where $d=$ density
$\begin{aligned} P_1 d_1^{-7} & =P_2 d_2^{-\gamma} \\ \frac{P_1}{P_2} & =\left(\frac{d_2}{d_1}\right)^{-\gamma} \Rightarrow \frac{P_1}{P_2}=\left(\frac{d_1}{d_2}\right)^\gamma \\ \frac{P}{P^{\prime}} & =\left(\frac{d}{d^{\prime}}\right)^{7 / 5} \Rightarrow \frac{P^{\prime}}{P}=\left(\frac{d^{\prime}}{d}\right)^{7 / 5} \\ & =(32)^{7 / 5} \\ & =\left(2^5\right)^{7 / 5}=2^7=128\end{aligned}$
For adiabatic process $P V^\gamma=$ constant where,
$V=$ volume
$\therefore \quad P d^{-7}=$ constant
$\left(\because v \propto \frac{1}{d}\right)$
where $d=$ density
$\begin{aligned} P_1 d_1^{-7} & =P_2 d_2^{-\gamma} \\ \frac{P_1}{P_2} & =\left(\frac{d_2}{d_1}\right)^{-\gamma} \Rightarrow \frac{P_1}{P_2}=\left(\frac{d_1}{d_2}\right)^\gamma \\ \frac{P}{P^{\prime}} & =\left(\frac{d}{d^{\prime}}\right)^{7 / 5} \Rightarrow \frac{P^{\prime}}{P}=\left(\frac{d^{\prime}}{d}\right)^{7 / 5} \\ & =(32)^{7 / 5} \\ & =\left(2^5\right)^{7 / 5}=2^7=128\end{aligned}$
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